The case of an N-gon

TL;DR

This paper constructs indefinite theta series for N-gons in certain symmetric spaces, proves their convergence, and interprets their coefficients as linking numbers, advancing understanding of mock modular forms in indefinite inner product spaces.

Contribution

It provides a new proof of the convergence of indefinite theta series associated with N-gons using linking numbers, avoiding explicit surface parametrization, and extends the method to higher signature cases.

Findings

Proved absolute convergence of the theta series.

Interpreted coefficients as linking numbers.

Extended the approach to higher signature spaces.

Abstract

We construct the indefinite theta series attached to N-gons in the symmetric space of an indefinite inner product space of signature (m-2,2) following the suggestions of section C in the recent paper of Alexandrov, Banerjee, Manschot, and Pioline. We prove the termwise absolute convergence of the holomorphic mock modular part of these series and also obtain an interpretation of the coefficients of this part as linking numbers. Thus we prove the convergence conjecture of ABMP provided none of the vectors in the collection CC={C_1,..., C_N} is a null vector. The use of linking numbers and a homotopy argument eliminates the need for an explicit parametrization of a surface S spanning the N-gon that was used in an essential way in our previous work. We indicate how our method could be carried over to a more general situation for signature (m-q,q) where higher homotopy groups are now involved. In the last section, we apply the method to the case of a dodecahedral cell in the symmetric space of a quadratic form of signature (m-3,3).